Euler Totient Research Articles

Symmetrical parallel RC Multiple-T Frequency Combinations

For a given number of resistance alms n, and capacitance arms nc in a symmetrical parallel RC multiple-T, it is possible to know the maximum number of dissimilar resonant frequency combinations theoretically from a formula derived using a combinational technique. Verification of the results is done by counting the number of dissimilar frequencies for the ratio nr/ nc. WM=WD (nrnc)1/2 and also from frequencies practically determined. The technique common to both these methods is the combinatorial nne as used in deriving Euler's Totient function in the theory of numbers.

ON SUMMATION OF SUBSERIES OF THE RIEMANN ZETA FUNCTION

Three classes of subseries of the Riemann zeta function are evalu- ated in closed form. All the results are expressed in terms of the Riemann zeta function itself and powers of �. By using the Euler totient function, we prove that these three classes are unique subseries that exhibit this kind of closed form.

Aesthetics and quality of numbers using the primety measure

The Farey sequences can be used to create the Euler totient function, by identifying the fractions for number n that did not occur in all Farey sequences up to n - 1. This function creates, when divided by n - 1, what is here called the primety measure, which is a measure of how close to being a prime number n is. The quality of a number sequence can be determined using the primety, which is further generalised to real numbers through the use of real numbered Farey sequences. Ambiguous and interesting results are obtained in this process; for instance it seems that all integers approach full primety when investigated in the real domain. The corresponding numerical sequences are furthermore shown to have interesting artistic properties. In particular the fractal behaviour is interesting. To further investigate fractal ambiguity, common fractals are extended with random elements, which render softer or more varied shapes.

GENERALIZED CULLEN NUMBERS WITH THE LEHMER PROPERTY

We say a positive integer n satisfies the Lehmer property if <TEX>${\phi}(n)$</TEX> divides n - 1, where <TEX>${\phi}(n)$</TEX> is the Euler's totient function. Clearly, every prime satisfies the Lehmer property. No composite integer satisfying the Lehmer property is known. In this article, we show that every composite integer of the form <TEX>$D_{p,n}=np^n+1$</TEX>, for a prime p and a positive integer n, or of the form <TEX>${\alpha}2^{\beta}+1$</TEX> for <TEX>${\alpha}{\leq}{\beta}$</TEX> does not satisfy the Lehmer property.

Omega theorems related to the general Euler totient function

We prove an omega estimate related to the general Euler totient function associated to a polynomial Euler product satisfying some natural analytic properties. For convenience, we work with a set of L-functions similar to the Selberg class, but in principle our results can be proved in a still more general setup. In a recent paper the authors treated a special case of Dirichlet L-functions with real characters. Greater generality of the present paper invites new technical difficulties. Effectiveness of the main theorem is illustrated by corollaries concerning Euler totient functions associated to the shifted Riemann zeta function, shifted Dirichlet L-functions and shifted L-functions of modular forms. Results are either of the same quality as the best known estimates or are entirely new.

Normal coverings and pairwise generation of finite alternating and symmetric groups

The normal covering number γ(G) of a finite, non-cyclic group G is the least number of proper subgroups such that each element of G lies in some conjugate of one of these subgroups. We prove that there is a positive constant c such that, for G a symmetric group Sym(n) or an alternating group Alt(n), γ(G)⩾cn. This improves results of the first two authors who had earlier proved that aφ(n)⩽γ(G)⩽2n/3, for some positive constant a, where φ is the Euler totient function. Bounds are also obtained for the maximum size κ(G) of a set X of conjugacy classes of G=Sym(n) or Alt(n) such that any pair of elements from distinct classes in X generates G, namely cn⩽κ(G)⩽2n/3.

Clique Dominating Sets of Euler Totient Cayley Graphs

Graph Theory has been realized as one of the most flourishing branches of modern Mathematics finding widest applications in all most all branches of Sciences, Social Sciences, Engineering, Computer Science, etc. Number Theory is one of the oldest branches of Mathematics, which inherited rich contributions from almost all greatest mathematicians, ancient and modern. Using the number theoretic function Euler totient function we have defined an Euler totient Cayley graph and in this paper we study the Clique domination parameters of Euler totient Cayley graphs.

On square values of the product of the Euler totient and sum of divisors functions

If $n$ is a positive integer such that $\phi (n)\sigma (n)=m^2$ for some positive integer $m$, then $m\le n$. We put $m=n-a$ and we study the positive integers $a$ arising in this way.

Möbius Polynomials

SummaryWe introduce the Möbius polynomial , which gives the number of aperiodic bracelets of length n with x possible types of gems, and therefore satisfies Mn(x) ≡ 0 for all (mod n) for all x ϵ 핫. We derive some key properties, analyze graphs in the complex plane, and then apply Möbius polynomials combinatorially to juggling patterns, irreducible polynomials over finite fields, and Euler's totient theorem.

Some Properties of Direct Product Graphs of Cayley Graphs with Arithmetic Graphs

Nathanson was the pioneer in introducing the concepts of Number Theory, particularly, the “Theory of Congruences” in Graph Theory, thus paving way for the emergence of a new class of graphs, namely “Arithmetic Graphs”. Cayley graphs are another class of graphs associated with the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes an Arithmetic graph. In this paper, we present some results related to basic properties of direct product graphs of Euler totient Cayley graphs with Arithmetic graph.

On a generalization of the Euler totient function

For a general polynomial Euler product F(s) we define the associated Euler totient function φ(n, F) and study its asymptotic properties. We prove that for F(s) belonging to certain subclass of the Selberg class of L-functions, the error term in the asymptotic formula for the sum of φ(n, F) over positive integers n ≤ x behaves typically as a linear function of x. We show also that for the Riemann zeta function the square mean value of the error term in question is minimal among all polynomial Euler products from the Selberg class, and that this property uniquely characterizes ζ(s).

Composed products and factors of cyclotomic polynomials over finite fields

Let q = p s be a power of a prime number p and let $${\mathbb {F}_{\rm q}}$$ be a finite field with q elements. This paper aims to demonstrate the utility and relation of composed products to other areas such as the factorization of cyclotomic polynomials, construction of irreducible polynomials, and linear recurrence sequences over $${\mathbb {F}_{\rm q}}$$ . In particular we obtain the explicit factorization of the cyclotomic polynomial $${\Phi_{2^nr}}$$ over $${\mathbb {F}_{\rm q}}$$ where both r ? 3 and q are odd, gcd(q, r) = 1, and $${n\in \mathbb{N}}$$ . Previously, only the special cases when r = 1, 3, 5, had been achieved. For this we make the assumption that the explicit factorization of $${\Phi_r}$$ over $${\mathbb {F}_{\rm q}}$$ is given to us as a known. Let $${n = p_1^{e_1}p_2^{e_2}\cdots p_s^{e_s}}$$ be the factorization of $${n \in \mathbb{N}}$$ into powers of distinct primes p i , 1 ? i ? s. In the case that the multiplicative orders of q modulo all these prime powers $${p_i^{e_i}}$$ are pairwise coprime, we show how to obtain the explicit factors of $${\Phi_{n}}$$ from the factors of each $${\Phi_{p_i^{e_i}}}$$ . We also demonstrate how to obtain the factorization of $${\Phi_{mn}}$$ from the factorization of $${\Phi_n}$$ when q is a primitive root modulo m and $${{\rm gcd}(m, n) = {\rm gcd}(\phi(m),{\rm ord}_n(q)) = 1.}$$ Here $${\phi}$$ is the Euler's totient function, and ord n (q) denotes the multiplicative order of q modulo n. Moreover, we present the construction of a new class of irreducible polynomials over $${\mathbb {F}_{\rm q}}$$ and generalize a result due to Varshamov (Soviet Math Dokl 29:334---336, 1984).

Cyclotomic Polynomials, Symmetric Polynomials, and a Generalization of Euler's Totient Function

SummaryWe introduce a generalization of Euler's totient function that, when applied to an integer n ≥ 2, can be written as a polynomial in the prime factors of n. We then show how cyclotomic and complete homogeneous symmetric polynomials appear as factors of these polynomials when n has at most two distinct prime divisors.

Cyclotomic Polynomials, Symmetric Polynomials, and a Generalization of Euler's Totient Function

Cyclotomic Polynomials, Symmetric Polynomials, and a Generalization of Euler's Totient Function

A Probabilistic Look at Series Involving Euler's Totient Function

Abstract.We use a probabilistic method to evaluate the limit of

Substitutions with Vanishing Rotationally Invariant First Cohomology

The cohomology groups of tiling spaces with three‐fold and nine‐fold symmetries are obtained. The substitution tilings are characterized by the fact that they have vanishing first cohomology group in the space of tilings modulo a rotation. The rank of the rational first cohomology, in the tiling space formed by the closure of a translational orbit, equals the Euler totient function evaluated at N if the underlying rotation group is ZN. When the symmetries are of crystallographic type, the cohomologies are infinitely generated.

The Least Prime Congruent to One Modulo n

It is known that there are infinitely many primes congruent to 1 (mod n) for any integer n > 1. In this paper, we use an elementary argument to prove that the least such prime is at most 2ϕ(n) + 1 −1, where ϕ is the Euler totient function.

Correlation of fractions with divisibility constraints

AbstractLet \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$ B=(B_{Q}\!)_{{Q \in {\mathbb N}}} $\end{document} be an increasing sequence of positive square free integers satisfying the condition that $ B_{{Q_1}}\vert B_{{Q_2}} $ whenever Q1 &lt; Q2. For any subinterval I ⊂ [0, 1], let It is shown that if BQ ≪ Qlog log Q/4, then the limiting pair correlation function of the sequence \documentclass{article}\usepackage{amsmath,amssymb,mathrsfs,bm}\pagestyle{empty}\begin{document}$ ({\mathscr{F}_{{B}\!,_Q}(I)})_{Q \in {\mathbb N}} $\end{document} exists and is independent of the subinterval I. Moreover, the sequence is Poissonian if $ \lim_{Q \rightarrow \infty }{{\varphi (B_{Q}\!)}\over{B_{Q}\!}} = 0 $, and exhibits a very strong repulsion if $ \lim_{Q \rightarrow \infty }{{\varphi (B_{Q}\!)}\over{B_{Q}\!}} \ne 0 $, where φ is Euler's totient function. © 2011 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim

On the distribution of the partial sum of Euler's totient function in residue classes

On the distribution of the partial sum of Euler's totient function in residue classes

Colourings of cyclotomic integers with class number one

This paper continues the study of colourings of the sets of cyclotomic integers ℳ n = ℤ[ξ n ] (ξ n = e 2πi/n , a primitive nth root of unity) with class number one. We present results for the colour symmetry group and colour preserving group for a given ideal colouring of ℳ n , with φ(n) = 8 and 10, thus completing the characterisation of the colour preserving group for the cases φ(n) ≤ 10, where φ is Euler's totient function.